Spectral Method for Phase Retrieval: An Expectation Propagation Perspective
Ma, Junjie1,2,3; Dudeja, Rishabh1; Xu, Ji4; Maleki, Arian1; Wang, Xiaodong2
AbstractPhase retrieval refers to the problem of recovering a signal x(star) is an element of C-n from its phaseless measurements y(i) = vertical bar a(i)(H) x(star)vertical bar, where {alpha(i)}(i=1)(m) are the measurement vectors. Spectral method is widely used for initialization in many phase retrieval algorithms. The quality of spectral initialization can have a major impact on the overall algorithm. In this paper, we focus on the model where A = [alpha(1), ... , alpha(m)](H) has orthonormal columns, and study the spectral initialization under the asymptotic setting m, n -> infinity with m/n infinity -> delta is an element of (1, infinity). We use the expectation propagation framework to characterize the performance of spectral initialization for Haar distributed matrices. Our numerical results confirm that the predictions of the EP method are accurate for not-only Haar distributed matrices, but also for realistic Fourier based models (e.g. the coded diffraction model). The main findings of this paper are the following: 1) There exists a threshold on delta(denoted as delta(weak)) below which the spectral method cannot produce a meaningful estimate. We show that delta(weak) = 2 for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari show that delta(weak) = 1 for the i.i.d. Gaussian model. 2) The optimal design for the spectral method coincides with that for the i.i.d. Gaussian model, where the latter was recently introduced by Luo, Alghamdi and Lu.
KeywordPhase measurement Message passing Tools Signal processing algorithms Prediction algorithms Numerical models Approximation algorithms Phase retrieval spectral method coded diffraction pattern expectation propagation (EP) approximate message passing (AMP) state evolution orthogonal AMP vector AMP
Indexed BySCI
Funding ProjectNational Science Foundation (NSF)[CCF 1814803] ; Office of Naval Research (ONR)[N000141712827]
WOS Research AreaComputer Science ; Engineering
WOS SubjectComputer Science, Information Systems ; Engineering, Electrical & Electronic
WOS IDWOS:000612137400036
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Document Type期刊论文
Corresponding AuthorMa, Junjie
Affiliation1.Columbia Univ, Dept Stat, New York, NY 10027 USA
2.Columbia Univ, Dept Elect Engn, New York, NY 10027 USA
3.Chinese Acad Sci, Inst Computat Math & Sci Engn Comp, Acad Math & Syst Sci, Beijing 100190, Peoples R China
4.Columbia Univ, Dept Comp Sci, New York, NY 10027 USA
Recommended Citation
GB/T 7714
Ma, Junjie,Dudeja, Rishabh,Xu, Ji,et al. Spectral Method for Phase Retrieval: An Expectation Propagation Perspective[J]. IEEE TRANSACTIONS ON INFORMATION THEORY,2021,67(2):1332-1355.
APA Ma, Junjie,Dudeja, Rishabh,Xu, Ji,Maleki, Arian,&Wang, Xiaodong.(2021).Spectral Method for Phase Retrieval: An Expectation Propagation Perspective.IEEE TRANSACTIONS ON INFORMATION THEORY,67(2),1332-1355.
MLA Ma, Junjie,et al."Spectral Method for Phase Retrieval: An Expectation Propagation Perspective".IEEE TRANSACTIONS ON INFORMATION THEORY 67.2(2021):1332-1355.
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